Search Results (12)

Jacobian-free Newton Krylov (JFNK) is an attractive method to solve nonlinear equations in the nuclear engineering community, and has been successfully applied to steady-state neutron diffusion k-eigenvalue problems and multi-physics coupling problems. Preconditioning technique plays an important role in the JFNK algorithm, significantly affecting its computational efficiency. The key point is how to automatically construct a high-quality preconditioning matrix that can improve the convergence rate and perform the preconditioning matrix factorization efficiently and robustly. A reordering-based ILU(k) preconditioner is proposed to achieve the above objectives. In detail, the finite difference technique combined with the coloring algorithm is utilized to automatically construct a preconditioning matrix with low computational cost. Furthermore, the reordering algorithm is employed for the ILU(k) to reduce the additional non-zero elements and pursue robust computational performance. A 2D LRA neutron steady-state benchmark problem is used to evaluate the performance of the proposed preconditioning technique, and a steady-state neutron diffusion k-eigenvalue problem with thermal-hydraulic feedback is also utilized as a supplement. The results show that coloring algorithms can automatically and efficiently construct the preconditioning matrix. The computational efficiency of the FDP with coloring could be about 60 times higher than that of the preconditioner without the coloring algorithm. The reordering-based ILU(k) preconditioner shows excellent robustness, avoiding the effect of the fill-in level k choice in incomplete LU factorization. Moreover, its performances under different fill-in levels are comparable to the optimal computational cost with natural ordering. Full article

Many probability-based uncertainty quantification (UQ) schemes require a large amount of sampled data to build credible probability density function (PDF) models for uncertain parameters. Unfortunately, the amounts of data collected as to compressor blades of aero-engines are mostly limited due to the expensive and time-consuming tests. In this paper, we develop a preconditioner-based data-driven polynomial chaos (PDDPC) method that can efficiently deal with uncertainty propagation of limited amounts of sampled data. The calculation accuracy of a PDDPC method is closely related to the sample size of collected data. Therefore, the influence of sample size on this PDDPC method is investigated using a nonlinear test function. Subsequently, we consider the real manufacturing errors in stagger angles for compressor blades. Under three different operating conditions, the PDDPC method is applied to investigate the effect of stagger-angle error on UQ results of multiple aerodynamic parameters of a two-dimensional compressor blade. The results show that as the sample-size of measured data increases, UQ results regarding aerodynamic performance obtained by the PDDPC method gradually converge. There exists a critical sample size that ensures accurate UQ analysis of compressor blades. The probability information contained in the machining error data is analyzed through Kullback–Leibler divergence, and the critical sample size is determined. The research results can serve as a valuable reference for the fast and cheap UQ analysis of compressor blades in practical engineering. Full article

The study of ship waves is important for ship detection, coastal erosion and wave drag. This paper proposed a highly paralleled numerical computation method for efficiently simulating three-dimensional nonlinear kelvin waves. First, a numerical model for nonlinear ship waves is established based on potential flow theory, the Jacobian-free Newton–Krylov (JFNK) method and the boundary integral method. To reduce the amount of data stored in the JFNK method and improve the computational efficiency, a banded preconditioner method is then developed by formulating the optimal bandwidth selection rule. After that, a Graphics Process Unit (GPU)-based parallel computing framework is designed, and we used the Compute Unified Device Architecture (CUDA) language to develop a GPU solution. Finally, numerical simulations of 3D nonlinear ship waves under multiple scales are performed by using the GPU and CPU solvers. Simulation results show that the proposed GPU solver is more efficient than the CPU solver with the same accuracy. More than 66% GPU memory can be saved, and the computational speed can be accelerated up to 20 times. Hence, the computation time for Kelvin ship waves simulation can be significantly reduced by applying the GPU parallel numerical scheme, which lays a solid foundation for practical ocean engineering. Full article

Total fractional-order variation (TFOV) in image deblurring problems can reduce/remove the staircase problems observed with the image deblurring technique by using the standard total variation (TV) model. However, the discretization of the Euler–Lagrange equations associated with the TFOV model generates a saddle point system of equations where the coefficient matrix of this system is dense and ill conditioned (it has a huge condition number). The ill-conditioned property leads to slowing of the convergence of any iterative method, such as Krylov subspace methods. One treatment for the slowness property is to apply the preconditioning technique. In this paper, we propose a block triangular preconditioner because we know that using the exact triangular preconditioner leads to a preconditioned matrix with exactly two distinct eigenvalues. This means that we need at most two iterations to converge to the exact solution. However, we cannot use the exact preconditioner because the Shur complement of our system is of the form $S=K^{*}K+\lambda L_{\alpha }$ which is a huge and dense matrix. The first matrix, $K{}^{*}K$, comes from the blurred operator, while the second one is from the TFOV regularization model. To overcome this difficulty, we propose two preconditioners based on the circulant and standard TV matrices. In our algorithm, we use the flexible preconditioned GMRES method for the outer iterations, the preconditioned conjugate gradient (PCG) method for the inner iterations, and the fixed point iteration (FPI) method to handle the nonlinearity. Fast convergence was found in the numerical results by using the proposed preconditioners. Full article

Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used in European option pricing; however, to our knowledge, American option pricing based on such models and their numerical methods is yet to be studied. Hence, a fast algorithm for solving the multi-state time-fractional linear complementary problem arising from the regime-switching time-fractional American option pricing models is developed in this paper. To construct the solution strategy, the original problem has been converted into a Hamilton–Jacobi–Bellman equation, and a nonlinear finite difference scheme has been proposed to discretize the problem with stability analysis. A policy-Krylov subspace method is developed to solve the nonlinear scheme. Further, to accelerate the convergence rate of the Krylov method, a tri-diagonal preconditioner is proposed with condition number analysis. Numerical experiments are presented to demonstrate the validity of the proposed nonlinear scheme and the efficiency of the proposed preconditioned policy-Krylov subspace method. Full article

This work aims to supplement the realization and validation of a higher-order well-balanced unstructured finite volume (FV) scheme, that has been relatively recently presented, for numerically simulating weakly non-linear weakly dispersive water waves over varying bathymetries. We investigate and develop solution strategies for the sparse linear system that appears during this FV discretisation of a set of extended Boussinesq-type equations on unstructured meshes. The resultant linear system of equations must be solved at each discrete time step as to recover the actual velocity field of the flow and advance in time. The system’s coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the undisturbed water depth and the mesh topology. To this end, we investigate the application of different well-known iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. The iiterative methods considered are the GMRES and the BiCGSTAB, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is finally proposed which combines the use of the BiCGSTAB method with the ILUT preconditioner and the Reverse Cuthill–McKee reordering. Full article

The principal objective of this work was to develop a semi-implicit hybrid boundary element method (HBEM) to describe the nonlinear fractional biomechanical interactions in functionally graded anisotropic (FGA) soft tissues. The local radial basis function collocation method (LRBFCM) and general boundary element method (GBEM) were used to solve the nonlinear fractional dual-phase-lag bioheat governing equation. The boundary element method (BEM) was then used to solve the poroelastic governing equation. To solve equations arising from boundary element discretization, an efficient partitioned semi-implicit coupling algorithm was implemented with the generalized modified shift-splitting (GMSS) preconditioners. The computational findings are presented graphically to display the influences of the graded parameter, fractional parameter, and anisotropic property on the bio-thermal stress. Different bioheat transfer models are presented to show the significant differences between the nonlinear bio-thermal stress distributions in functionally graded anisotropic biological tissues. Numerical findings verified the validity, accuracy, and efficiency of the proposed method. Full article

The ship wave is of great interest for wave drag and coastal erosion. This paper proposes a mechanism of ship wave transformation to explore the effects of ship speed and ship size on the waveform. Firstly, based on the theory of potential flow, the boundary integral equations for the Kelvin ship waves are obtained by deploying the different Kelvin sources or Rankine sources. Then, these integral equations are numerically discretized to a set of nonlinear equations. Finally, the Jacobian−free Newton–Krylov method with a preconditioner is adopted to solve the nonlinear equations. Though imitating plenty of different Kelvin wave patterns, the mechanism of ship wave transformation is proposed to conveniently generate the polymorphic Kelvin wake patterns. The above numerical simulation scheme is verified by comparing simulation results with real ship waves. After that, the wake angle is discussed with the effects of Froude number, source strength and source type by following the mechanism of ship wave transformation. The results show that the wake angle tends to decrease with ship speed but increase with ship size. In addition, for high ship speeds, the effect on the wake angle can be more dramatic. Full article

We consider the finite element approximation of the solution to elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. We propose a new nonlinear version of preconditioning, dedicated to nonlinear substructured and condensed formulations with dual approach, i.e., nonlinear analogues to the Finite Element Tearing and Interconnecting (FETI) solver. By increasing the importance of local nonlinear operations, this new technique reduces communications between processors throughout the parallel solving process. Moreover, the tangent systems produced at each step still have the exact shape of classically preconditioned linear FETI problems, which makes the tractability of the implementation barely modified. The efficiency of this new preconditioner is illustrated on two academic test cases, namely a water diffusion problem and a nonlinear thermal behavior. Full article

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems $A_k{\mathbf{x}}_k={\mathbf{b}}_k,k=1,\dots$ arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates. Full article

To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) iteration method for solving a class of large sparse and complex symmetric weakly nonlinear equations. At each inner process iterative step of the new methods, one only needs to solve the real subsystems with the same symmetric positive and definite coefficient matrix. Therefore, the computational workloads and computational storage will be saved in actual implements. The conditions for guaranteeing the local convergence are studied in detail. The quasi-optimal parameters are also proposed for both the C-to-R-based Picard iteration method and the NC-to-R iteration method. Numerical experiments are performed to show the efficiency of the new methods. Full article

A modification to an existing iterative method for computing zeros with unknown multiplicities of nonlinear equations or a system of nonlinear equations is presented. We introduce preconditioners to nonlinear equations or a system of nonlinear equations and their corresponding Jacobians. The inclusion of preconditioners provides numerical stability and accuracy. The different selection of preconditioner offers a family of iterative methods. We modified an existing method in a way that we do not alter its inherited quadratic convergence. Numerical simulations confirm the quadratic convergence of the preconditioned iterative method. The influence of preconditioners is clearly reflected in the numerically achieved accuracy of computed solutions. Full article